\( \left. \begin{array} { l } { - x + y + 3 z = 4 } \\ { 3 x + y + 2 z = 1 } \\ { - 2 x - 2 y \pm z = 3 } \end{array} \right. \)

To solve this system of equations, you can first rewrite it in matrix form or use substitution/elimination methods to find the values of \( x \), \( y \), and \( z \). These equations represent planes in three-dimensional space, and their solution, if it exists, will be the point where all three planes intersect. One common mistake people make is forgetting to check for inconsistencies in the equations. For example, if you end up with a false statement (like \(0 = 1\)), it means the planes do not intersect at a single point, possibly being parallel or coinciding in some manner! Remember to check your calculations carefully! Another useful tip is to simplify the system by isolating one variable, substituting back into the other equations, and reducing them step by step; that way, you not only clarify each part but also break down the complexity of the system into manageable pieces! Happy solving!